Rigid symmetrical tops

Show that the rotational energy of a rigid symmetric top is given by... 

M. M. Tirado and J. Garcia de la Torre, Rotational dynamics of rigid symmetric top molecules - Application to circular cylinders, J. Chem. Phys., 73 (1980) 1986-1993. 

We will presently give the explicit expressions for the /> and the ( pj pX y for the linear momenta of the translational motion of the center of mass. The corresponding expressions for the rotational motion will be determined separately for the case of rigid linear rotators and rigid symmetric tops in the next sections. 

In Section III we have found a formal expression for the distribution function with accuracy up to terms in P. By an analogous procedure we will presently determine the expression of the distribution function correct to order P, for a system of interacting rigid symmetric top molecules. 

Now consider rotational diffusion of a rigid symmetric top in an oriented environment such as a lipid bilayer or other liquid crystal. The top (e.g., cylinder) could represent one of the lipids, a probe, or a protein. It may be useful to think of the vector wobbling in a cone , as sketched in Figure 5. The correlation function clearly will decay to a nonzero plateau value, because angular averaging is incomplete. One may also anticipate that the relaxation will be more rapid than free diffusion due to effects of the ordering potential. 

Tirado M.M. and Garcia de la Torre J. 1979. Translational friction coefficients of rigid, symmetric top macromolecules. Application to circular cylinders. /. Chem. Phys. 71(6) 2581-2587. 

In the previous section we connected the susceptibility tensor directly to the microscopic parameters, the molecular polarizabilities. Indeed, this point of view clarifies the molecular aspects of the x tensor, but the connection with the dynamical model is complicated by the extremely large number of variables that must be considered. An alternative possibility is represented by a coarsegrained definition of the susceptibility function, in fact if we disregard to the microscopic information we gain a more direct link to the dynamic model. The polarizability of a rigid symmetric-top molecule can be described by (2.29), hence the dielectric tensor becomes... 

A small step rotational diffusion model has been used to describe molecular rotations (MR) of rigid molecules in the presence of a potential of mean torque.118 120,151 t0 calculate the orientation correlation functions, the rotational diffusion equation must be solved to give the conditional probability for the molecule in a certain orientation at time t given that it has a different orientation at t = 0, and the equilibrium probability for finding the molecule with a certain orientation. These orientation correlation functions were found as a sum of decaying exponentials.120 In the notation of Tarroni and Zannoni,123 the spectral denisities (m = 0, 1, 2) for a deuteron fixed on a reorienting symmetric top molecule are ... 

The spherical pendulum, which consists of a mass attached by a massless rigid rod to a frictionless universal joint, exhibits complicated motion combining vertical oscillations similar to those of the simple pendulum, whose motion is constrained to a vertical plane, with rotation in a horizontal plane. Chaos in this system was first observed over 100 years ago by Webster [2] and the details of the motion discussed at length by Whittaker [3] and Pars [4]. All aspects of its possible motion are covered by the case, when the mass is projected with a horizontal speed V in a horizontal direction perpendicular to the vertical plane containing the initial position of the pendulum when it makes some acute angle with the downward vertical direction. In many respects, the motion is similar to that of the symmetric top with one point fixed, which has been studied ad nauseum by many of the early heroes of quantum mechanics [5]. 

Figure 7. (a) Concept of time-dependent alignment as a method for structural determination. Top Initial alignment at t = 0, dephasing, and recurrence of alignment at later times. Bottom Classical motion of a rigid prolate symmetric top. (b) Structures of stilbene and tryptamine-water complex from rotational coherence spectroscopy transients are shown, [see ref. 13]. 

When the overall motion is not isotropic, the diagonal elements of the rotational diffusion tensor are no longer equivalent and rotation about the three principal axes of the diffusion tensor may be described by different diffusion coefficients or correlation times. For anisotropic motion, the correlation time in Eqs. 16 and 25 is an effective correlation time, r ff, containing contributions from the various modes of reorientation. Partitioning of the various components of rff can be achieved through appropriate dynamic models. The simplest case of anisotropic motion is that for a symmetric-top molecule. The r ff of a rigid ellipsoid is expressed in terms of two parameters, Dn and DL these two parameters respectively describe the rotational diffusion about the C3 symmetry axis (major axis) and the two perpendicular axes (minor axes), which are assumed to be equivalent25-44 (Fig. 4) ... 

From Eqs. (3.40) and (3.35) it is obvious that the inversion—rotation wave functions i//°. (0,, X, p) of NH3 which are the eigenfunctions of the operator, , can be written as a product of the rigid-rotor symmetric top wave functions depending on the Euler angles 0,4>, x and the inversion wave functions, depending on the variable p. Integration of the Schrodinger equation... 

Before discussing the spherical rotor, it is appropriate to focus on the rotational energy levels for symmetric rotors (see also Tab. 4.3-2). For the rigid prolate top, the rotational energy is given by... 

The approach for rigid rotators proposed above can be extended [90] to the orientational relaxation of an assembly of dipolar nonpolarizable symmetrical top molecules undergoing fractional diffusion in space (treated originally by McConnell [41], Morita [91], and Coffey et al. [8,92] for normal diffusion). The rotational Brownian motion of a symmetric top molecule in the molecular coordinate system oxyz rigidly connected to the top is characterized by the angular velocity co and the angular momentum M defined as [41]... 

Since we have considered only assemblages of point particles heretofore, we have not given the rules for setting up the wave equation for a rigid body. We shall not discuss these rules here1 but shall take the wave equation for the symmetrical top... 

Since the dynamics of rigid bodies is based on the dynamics of particles, these rules must be related to the rules given in Chapter IV. For a discussion of a method of finding the wave equation for a system whose Hamiltonian is not expressed in Cartesian coordinates, see B. Podolsky, Phys. Rev. 32, 812 (1928), and for the specific application to the symmetrical top see the references below. 

The Symmetrical Top.—If Ax denotes the moment of inertia about an axis perpendicular to the axis of symmetry (z), At the moment of inertia about the axis of symmetry, and d, dy, d, the components of the angular velocity in the system of reference (a , y, z) rigidly fixed in the body, then... 

Microcanonical transition-state theory (TST) assumes that all vibrational-rotational levels for the degrees of freedom orthogonal to the reaction coordinate have equal probabilities of being populated [12]. The quasi-classical normal-mode/rigid-rotor model described above may be used to choose Cartesian coordinates and momenta for these energy levels. Assuming a symmetric top system, the TS energy E is written as... 

To determine the quantum mechanical rigid-rotor energy levels, the quantum mechanical Hamiltonian operator is formed from the classical Hamiltonian in Eq. (2.50) and the eigenvalue equation, Eq. (2.54), is solved. For a symmetric top rigid-rotor, which has two equal moments of inertia (i.e., 4 = /(, /g), the resulting energy levels are... 

The Lagrange case. The case is characterized by the fact that A = B and 0 = yo = 0. This means that we deal with an axisymmetric rigid body, and on account of this, the Lagrange case is usually called the case of symmetric top. In other words, the symmetric top (or the Lagrange top) is a rigid body whose inertia ellipsoid is an ellipsoid of rotation and whose centre of gravity lies on the rotation axis (Fig. 4). 

For a cylindrically symmetric rigid body (a symmetric top) undergoing rotational diffusion, the distribution of reorientation angle 6 is given by [38-41]... 

Symmetric Tops attached to a Rigid Frame.—To a good approximation the rotational partition function of a molecule having several symmetric tops attached to a rigid frame may be expressed by " ... 

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